But this is a big paper, and I also want to talk about other papers, so I certainly won’t explain everything in here—just enough to help you get started! If you have questions, don’t be shy about asking them.

I thank Blake Pollard for filming this seminar, and Muhammad “Siddiq” Siddiqui-Ali for providing the videocamera and technical support.

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16 Responses to Network Theory Seminar (Part 1)

Just finished watching the video. I’m very impressed with both the quality and the work of the camera operator following you around. (Also the ambidextrous erase old stuff with one hand and write new stuff at the same time with the other is a neat trick.) Thanks to everyone involved in making this stuff available.

A bit of a silly question: so given the existence of superconductivity (ie, real things that do have 0 resistance), how does that relate to your statement that bad things will happen when you allow 0 for one of your edge labels?

I’m sure Blake will be pleased by your comment on his camera skills. I hadn’t even known erasing with one had while writing with the other was noteworthy—though now that you mention it, it’s easier if you’re left-handed, since we write left to right.

A bit of a silly question: so given the existence of superconductivity (ie, real things that do have 0 resistance), how does that relate to your statement that bad things will happen when you allow 0 for one of your edge labels?

That’s a fun physics question! Clearly a superconductor does not deliver infinite current when you put a finite voltage across it. So the question is: in what sense does it fail to have zero conductivity, and why… and why do people say it does have zero conductivity.

There are probably a number of limiting factors. Electrons can’t go faster than light, for one thing!

There are at least 23 different things called ‘graphs’, depending on whether we

• put arrows on the edges or not
• allow multiple edges from one node to another or not
• allow edges from a node to itself or not

The logically simplest kind of graph turns out to be the one that says “yes” to all these questions: it’s a directed multigraph

which is just a pair of functions from one set to another. I say it’s “logically simplest” because this concept doesn’t really use anything about sets: we can define a directed multigraph in any category as a pair of morphisms

from some object to some object So, when we start doing fancy stuff, it will easiest to work with this kind of graph.

I should note also that any category gives a graph of this kind, where is the set of morphisms and is the set of objects. We definitely want morphisms to have a direction, and to allow more than one morphism from one object to another, and to allow morphisms from an object to itself.

For Markov processes, we don’t really need multiple edges from one node to another, or edges from a node to itself. So, none of this matters much now except that the edges must have arrows on them. But the philosophy I prefer is one where we keep things simple by not going out of our way to exclude options.

To some people this simplicity seems like complexity, but if you ever get brainwashed by category theory you’ll agree that a pair of functions

is a simpler thing than a subset

To a category theorist, the subset is a pair of functions obeying the additional condition that is one-to-one: that is, an edge is uniquely determined by its source and target. This brings in the concept of ‘one-to-one’, or ‘monic’, making things a bit more elaborate.

Furthermore, I may sometimes want to have two roads from one city to another. So I don’t want my default notion of graph, on which I found network theory, to exclude this.

How mathematicians have different perspectives on the same object! For me is (or maybe was) an abstract data type. You just tell me the properties it has and the operations I am allowed to perform. How it is constructed does not interest me the slightest ;-) Like with real numbers.

You just tell me the properties it has and the operations I am allowed to perform. How it is constructed does not interest me the slightest ;-) Like with real numbers.

That’s an interesting example. In the category of sets the construction of the reals using Cauchy sequences gives a result isomorphic to the construction using Dedekind cuts. But in other topoi (i.e., other categories very much like the category of sets) the Cauchy reals and the Dedekind reals can differ!

I’ve never had any need to work with real numbers in a category other than the category of sets, so this seems rather abstruse to me. However, experts tell me that the Dedekind reals behave better.

I agree completely that when working with a given structure what matters is what we can do with it, not how it was built. This is the ‘structural’ attitude that category theorists love.

One reason it pays to analyze how a structure is built is that this gives clues on when we’ll be able to build analogous structures in other contexts.

From what I said, the concept of graph that I’m using in this course will generalize from the category of sets to any category whatsoever. The kind of graph you’re talking about will generalize to any category with binary products and reasonably well-behaved subobjects.

You see, in any category we can talk about

but it takes more bells and whistles to talk about

Here we are talking about subobjects () and binary products (). There are still tons of categories that have nicely behaved subobjects and binary products, so that’s still a huge class of contexts, but I like to leave my options open as long as possible. Of course if there were some reason I needed to use graphs of the sort I would do it in an instant.

For the categorically minded:

Puzzle. Using the former concept of graph, what’s a graph in the category of vector spaces? More precisely, what well-known category is equivalent to the category of graphs in the category of vector spaces?

By the way, Uwe: once upon a time we were talking about quantropy and I sort of discouraged you from studying some questions because I was feeling possessive about this idea and had ambitions of studying those questions myself. I apologize. That kind of possessiveness is almost always a dumb idea, and it was dumb in this case. I wound up getting too busy with other things to follow through on some of the questions you were starting to study!

I know in principle, but sometimes forget in practice, that it’s almost always better to encourage everyone to join in studying whatever I happen to be studying. At times someone will make so much progress that my own efforts become unnecessary. While this can be ego-damaging (and at times it’s felt terrible), the solution is for me to go do something else.

I read your pamphlets for nearly twenty years now. (Can that be true? Checking your Wiki page tells me that you started to write in 1993 which was about when I finished my phd. So yes, we are that old ;-) )

They were always **very inspiring** and **motivating** and I very much hope you **keep them coming**.

The quantropy series I liked a lot and did indeed some work on it. During that time I convinced myself that I need some serious knowledge of quantum field theory to progress further. That was beyond my resources and I shelved the project. I still think (maybe wishfull thinking) that with the ideas you have developed one can probably determine a minimal length scale for concrete quantum systems.

I think that your quantropy series was a prequel to what later became polymath projects. Your intention was different, but at least that was how I perceived it.

Yes, we’re getting old. The main problem with that (for me, right now) is that I feel unmotivated to re-explain things that I explained with great excitement 10 or 20 years ago. I think some people may wonder why I’m working on electrical circuits now. If I had the energy, I’d start with an explanation of why n-categories are so great, how they’re going to revolutionize our understanding of math and physics, and why we need to understand complex systems… and then I’d start talking about how to use n-categories (for n = 1 and 2) to think about electrical engineering in new ways. But I’m tired of giving the sales pitch—I want to just dive in and do the real work.

If you ever want to talk more about quantropy, let me know. Blake and I haven’t gotten around to publishing our paper on that, and while I think they’re some really exciting things left to understand, he’s moved on to other projects.

I wonder if you could say a little more about the “big picture” of network theory. What do you think the world needs a new kind of math for?

More specifically, you’ve talked about how network theory lets the different scientific disciplines “talk to each other”. How far do you think that goes? What role do you see network theory playing in relation to the project of scientific unification?

When you talk about unification with physicists they usually mean a GUT of some sort, but I mean ‘unification’ in the more general sense used in the philosophy of science (See: http://goo.gl/oTyD28)

Basically I’d like to know how systematically you think network theory can be applied across the sciences, and what implications (if any) it might have for the debates in the foundations of the sciences, for instance, concerning the autonomy of the special sciences (http://goo.gl/8F55lv).

I know of places where you’ve expressed skepticism at the too-quick application of physical theory to human behavior (http://goo.gl/y96cd6), and in general you’ve been very careful in your examples and applications. I’m certainly not asking you to say more than careful science allows. Still, papers like your Rosetta Stone (http://goo.gl/Dr8if) seem to hint at something more profound than merely easing communication. I wonder to what extent you explicitly see this work as contributing to the project of unification.

Thank you!

I replied:

Thanks for the questions! I have trouble talking about these issues at a high level of generality because I’ve been fighting it out in the trenches trying to understand many kinds of networks used in many different branches of science and engineering, trying to describe them in the language of modern mathematics, and trying to use this language to see how they fit together… this is a rather detailed project that’s just getting started; my overall view of it is much better than it was a year or two ago, but it still needs a lot of work.

The reason the world might need this new kind of math is something like this:

Scientists and engineers are struggling to understand complex systems made of interacting parts, and a lot of them feel frustrated: they’ve got lots of data and don’t feel good enough at extracting meaning from it. Biologists in particular tend to say they want new mathematical tools.

On the other hand, mathematicians have recently developed some powerful, simple tools for describing networks in a unified way – especially things like symmetric monoidal categories and operads. But these tools are not known to most scientists and engineers.

Conversely, the mathematicians who know these tools tend not to think much about biology, chemistry, electrical engineering and other ‘practical’ sciences where complex systems made of interacting parts show up. They’re much more likely to be thinking about string theory or pure math.

So, when it comes to network theory, a lot of branches of science and engineering have been left to develop their own mathematical tools without much interaction with the mathematicians who can think about networks in a very general way.﻿

On the other hand, mathematicians have recently developed some powerful, simple tools for describing networks in a unified way – especially things like symmetric monoidal categories and operads.

I would quibble with that. I would not gainsay that symmetric monoidal categories (SMCs) and operads are powereful. I am less sure that they are simple. And I am even less sure that they can tell you a lot about networks all by themselves. But perhaps you didn’t mean that.

If I were to go out on a limb, I would say there is unlikely to be one overarching theory of networks. Rather will end up with many different theories, which pay attention/model different aspects of networks. For example dynamics of networks (the network is evolving) is quite different from the dynamics on networks (a network models a dynamical system made up of interacting parts) and I have not seen anyone successfully combining these two perspectives.

The one thing I would heartily agree with John is that there is a lot of interesting mathematics carried out by scientists and engineers that applied mathematicians are largely unaware of (forget about pure). The most glaring example is hybrid dynamical systems, but there is plenty more.

Symmetric monoidal categories and operads are absurdly simple. Compare a full definition of the real numbers to a full definition of these other structures—it’s at least as long, and it’s a weird mix of ideas: algebraic structures, ordering and topology. I know because I’m teaching real analysis. “A complete ordered field” sounds short until you try to explain it from scratch… and it takes a lot of infrastructure—set theory, or second-order logic—to make sense of completeness: you’re quantifying over subsets. It takes quite a while to even construct an example!

The real numbers seem simple only because they’re familiar. We’ve all been forced to study them in school for many years… and the few people who don’t wind up failing, or hating math, find the real numbers to be simple and natural. It’s mainly just natural selection at work.

To me, category theory is less familiar but much more simple: it’s talking about the primitive abstract essences in their unadorned crystalline beauty. So in this sense (to revert to a previous conversation) I guess I am a category theorist. I really can’t think of any branch of math that’s simpler and clearer that category theory!

And I am even less sure that they can tell you a lot about networks all by themselves.

No, just as the real numbers all by themselves don’t say much about anything in science. These mathematical concepts are crucial if you want to say what’s going on, but you still need to have something to say.

If I were to go out on a limb, I would say there is unlikely to be one overarching theory of networks.

There will be a network of theories of networks… and someday, a theory of that network.

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